Solve for v w^2-v^2=u^2

$w^{2} - v^{2} = u^{2}$

Step 1

Subtract $w^{2}$ from both sides of the equation.

$- v^{2} = u^{2} - w^{2}$

Step 2

Divide each term in $- v^{2} = u^{2} - w^{2}$ by $- 1$ and simplify.

Divide each term in $- v^{2} = u^{2} - w^{2}$ by $- 1$ .

$\frac{- v^{2}}{- 1} = \frac{u^{2}}{- 1} + \frac{- w^{2}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{v^{2}}{1} = \frac{u^{2}}{- 1} + \frac{- w^{2}}{- 1}$

Divide $v^{2}$ by $1$ .

$v^{2} = \frac{u^{2}}{- 1} + \frac{- w^{2}}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{u^{2}}{- 1}$ .

$v^{2} = - 1 \cdot u^{2} + \frac{- w^{2}}{- 1}$

Rewrite $- 1 \cdot u^{2}$ as $- u^{2}$ .

$v^{2} = - u^{2} + \frac{- w^{2}}{- 1}$

Dividing two negative values results in a positive value.

$v^{2} = - u^{2} + \frac{w^{2}}{1}$

Divide $w^{2}$ by $1$ .

$v^{2} = - u^{2} + w^{2}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$v = \pm \sqrt{- u^{2} + w^{2}}$

Step 4

Simplify $\pm \sqrt{- u^{2} + w^{2}}$ .

Reorder $- u^{2}$ and $w^{2}$ .

$v = \pm \sqrt{w^{2} - u^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = w$ and $b = u$ .

$v = \pm \sqrt{(w + u) (w - u)}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$v = \sqrt{(w + u) (w - u)}$

Next, use the negative value of the $\pm$ to find the second solution.

$v = - \sqrt{(w + u) (w - u)}$

The complete solution is the result of both the positive and negative portions of the solution.

$v = \sqrt{(w + u) (w - u)}$

$v = - \sqrt{(w + u) (w - u)}$

$v = \sqrt{(w + u) (w - u)}$

$v = - \sqrt{(w + u) (w - u)}$